Here are solutions for each of the questions visible in the uploaded image:

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Question 28

Problem:
If $\frac{3}{4}$ of $\frac{1}{3}$ of $\frac{1}{5}$ of a number is 25, what is the number?

Solution:
Let the number be $x$. Then: $\frac{3}{4} \cdot \frac{1}{3} \cdot \frac{1}{5} \cdot x = 25$ Simplify the left-hand side: $\frac{1}{20} \cdot x = 25$ Multiply both sides by 20: $x = 500$ Answer: (D) 500

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Question 29

Problem:
Find the remainder when $17^{200}$ is divided by 18.

Solution:
Using Fermat's Little Theorem: a^{p-1} \equiv 1 \mod p \text{ (when a$and$p are coprime)} Here, $17$ and $18$ are coprime. Since $p = 18$, $17^{\phi(18)} \equiv 1 \mod 18$.
$\phi(18) = 6$, so $17^6 \equiv 1 \mod 18$.

Now, divide $200$ by $6$: $200 = 6 \times 33 + 2$ Thus, $17^{200} \equiv 17^2 \mod 18$. Calculate $17^2 \mod 18$: $17^2 = 289 \quad \text{and} \quad 289 \div 18 = 16 \text{ remainder } 1$ So, the remainder is $1$.

Answer: (A) 1

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Question 30

Problem:
A number, when divided by 296, leaves a remainder of 75. When the same number is divided by 37, what is the remainder?

Solution:
Using modular arithmetic: $x \equiv 75 \mod 296$ We know $296 = 37 \times 8$. Since $x$ modulo 296 also satisfies modulo 37: $75 \div 37 = 2 \text{ remainder } 1$ So, the remainder is $1$.

Answer: (A) 1

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Question 31

Problem:
When a natural number $n$ is divided by 4, the remainder is 3. What is the remainder when $2n$ is divided by 4?

Solution:
If $n \equiv 3 \mod 4$, then: $2n \equiv 2 \cdot 3 \mod 4$ $2n \equiv 6 \mod 4 \quad \Rightarrow \quad 6 \div 4 = 1 \text{ remainder } 2$ So, the remainder is $2$.

Answer: (B) 2

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Question 32

Problem:
A man traveled $\frac{2}{11}$ of his journey by coach, $\frac{7}{11}$ by rail, and walked the remaining 1 km. How far did he go?

Solution:
Let the total distance be $x$. Then: $\frac{2}{11}x + \frac{7}{11}x + 1 = x$ Simplify: $\frac{9}{11}x + 1 = x$ $1 = \frac{2}{11}x \quad \Rightarrow \quad x = \frac{11}{2} \times 1 = 11$ The total distance is $11$ km.

Answer: (A) 22 km

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Question 33

Problem:
A boy was asked to multiply a number by 53. By mistake, he multiplied it by 35. If his answer was 1206 less than the correct one, find the number.

Solution:
Let the number be $x$. The equation is: $53x - 35x = 1206$ Simplify: $18x = 1206 \quad \Rightarrow \quad x = \frac{1206}{18} = 67$ Answer: (B) 67

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Summary of Answers:

28. (D) 500
29. (A) 1
30. (A) 1
31. (B) 2
32. (A) 22 km
33. (B) 67

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Would you like me to explain any of the solutions in more detail?
Here are five related questions to extend your understanding:

1. What is Fermat's Little Theorem and how is it used in modular arithmetic problems?
2. How do you simplify fractional equations like in Question 28?
3. Why is modular arithmetic useful in remainder problems like Question 30?
4. Can you generalize the steps for solving problems like Question 32?
5. What are practical tips to avoid mistakes in word-based arithmetic problems?

Tip: Always verify the final step of your calculations, especially when working with modular arithmetic!